Visser only came on after Webster came off after 15 minutes with a rib complaint, but team-mate Craig Hamilton had more to worry about after requiring hospital treatment following a second-half head injury.
Scrum V analysis: Edinburgh 33-17 Ospreys
The home side dominated the early exchanges but struggled to find their way through the Ospreys defence.
However, Cairns finally broke the deadlock, going over after a one-two with Visser drew two defenders with David Blair converting.
Ospreys went close to scoring as the first half drew to a close, but full-back Gareth Owen lacked support following a promising opening.
Four quick-fire second-half tries quickly took the game beyond Ospreys, however.
Cairns and Visser reversed roles for the first as Cairns broke from deep before passing to Dutchman Visser running in from the left to score.
Visser then took possession of the ball on his 10-metre line on the right wing before he kicked long and chased the ball over.
Hooker Andrew Kelly then fed Visser and the winger ran in under the posts.
Visser's counterpart on the other wing, Mark Robertson then ran in a try from his own half, with replacement Phil Godman kicking over all but one conversion.
TV evidence got the visitors on the scoresheet when winger Nikki Walker was adjudged to have gone over, but the score stayed unconverted.
Jerry Collins and Sonny Parker scored late tries for the visitors, but the Edinburgh defence held firm to deny Ospreys a bonus point.
Ospreys scrum-half Mike Phillips: "We're very disappointed, we made too many errors and they capitalised.
Mike Phillips says he struggled after taking an early knock
"Our defence was everywhere, we let ourselves down badly.
"We were full of confidence coming up here and had a great side out, but they stopped us playing with three sevens in the back row.
"We're a lot better side than that and have to push on from here.
"I took a bit of a knock in the first minute that was really annoying, but it was great to be back on the field and playing rugby."
Edinburgh: Cairns, Robertson, De Luca, Houston, Webster, D. Blair, M. Blair, Traynor, Kelly, G. Cross, MacLeod, S. Turnbull, MacDonald, Rennie, R. Grant, C. Hamilton. Replacements: Visser for Webster (15), Godman for D. Blair (40), Laidlaw for M. Blair (40), C. Hamilton for MacLeod (57), Crombie for C. Hamilton (65). Not Used: Callam.
Ospreys: Owen, Vaughton, Parker, Bishop, Walker, Biggar, M. Phillips, Bevington, Shervington, Mitchell, Gough, Evans, Collins, Holah, Tiatia. Replacements: Nutbrown for M. Phillips (51), Breeze for Shervington (57), Griffiths for Mitchell (57), Tandy for Tiatia (47). Not Used: Goode, Beck, B. Davies.
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